Random Hermitian versus random non-Hermitian operators—unexpected links

We present some unexpected links between the spectral properties of ensembles of large Hermitian and non-Hermitian random matrices, within the formalism of free random variables. We address the task of adding non-Hermitian random matrices. To solve this problem, we present a new approach [1] based on a generalization of the notion of the matrix-valued Green's function to the quaternion level, and on an extension of free probability theory, which allows us to introduce the setting of quaternion-valued free probability theory. We use this quaternion construction to solve the problem of a non-Hermitian random matrix of the type H1 + iH2, with Hermitian H1, H2 freely independent. Finally, we mention conformal mapping relating spectral properties of Hermitian H1 + H2 to spectral properties of the non-Hermitian H1 + iH2 model.

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